I am learning about direct product of families of sets in a category. However, in my notes it states something along the lines of '...assuming representability of the direct product $\prod_{j \in J} A_j$ of a family of objects...'
What is a representation of a direct product? Is there some closely related functor that can describe this direct product? Possibly even $\prod_{j \in J} Hom(- , A_j)?$
Yes, one way to describe the direct product $\prod_{j \in J} A_j$ is precisely that it is the object representing the functor $\prod_{j \in J} \text{Hom}(-, A_j)$, if it exists. You can make a similar statement about arbitrary limits and colimits.